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They say 1-1+1-1+1-1+... is 1/2 (Grandi's Series fully explained)

Math tutorial answering the question from Reddit, why 1-1+1-1+1-1+... is 1/2? This is known as the Grandi's series. This question is from Reddit r/homeworkhelp https://www.reddit.com/r/HomeworkHelp/s/mG2OPxABKx Related links 1+2+3+...=-1/12 by@numberphile https://youtu.be/w-I6XTVZXww?si=9aEG3INL9delT4LI Cesàro Summation: https://en.wikipedia.org/wiki/Ces%C3%A0ro_summation Ramanujan Summation: https://en.wikipedia.org/wiki/Ramanujan_summation Cauchy Principal Value: https://en.wikipedia.org/wiki/Cauchy_principal_value 🛍 Shop my math t-shirts & hoodies: 👉 https://amzn.to/3qBeuw6 👉 Support this channel and get more content https://www.patreon.com/blackpenredpen #blackpenredpen #calculus

Grandi's Series Cesàro Summation Ramanujan Summation

4 weeks ago·youtube.com

+11

f(5)=? American Invitational Mathematics Examination

We will solve a hard functional equation problem from the 2012 American Invitational Mathematics Examination (AIME) . Given f(0)=1, f(x)*f(2x^2)=f(2x^3+x), and f(2)+f(3)=125, we need to find f(5). This is a question asked by one of my viewers! 🛍 Shop my math t-shirts & hoodies: 👉 https://amzn.to/3qBeuw6 👉 Support this channel and get more content https://www.patreon.com/blackpenredpen #blackpenredpen #calculus

2012 American Invitational Mathematics Examination

1 month ago·youtube.com

A new "all-in-one" calculus problem (now in a matrix)

I crafted another all-in-one calculus problem, and this time I put the 4 main ingredients of calculus, i.e. limit, derivative, integral, and series, into a 2 by 2 matrix. Oh btw, we will also have to use the Lambert W function. #blackpenredpen #math

Calculus Problem Solving Lambert W Function

1 month ago·youtube.com

+12

Mathematician Attempts Art - Numberphile

Hannah Fry returns to Numberphile to showcase her art skills (and show us her new portrait at The Royal Society). See the Objectivity video (for more about the REAL artwork) at https://youtu.be/3HLxB5LbrDU More links & stuff in full description below ↓↓↓ (Real) artist Chloe Barnes can be found at: https://www.chloebarnesartist.com/ (and a handful of limited edition prints of her Hannah portrait were still available at the time we published this video!) Thanks also to artist Paulina Kwietniewska who did the painting of Chloe: https://paulinakwietniewska.com/ The Royal Society collections: https://royalsociety.org/about-us/what-we-do/collections/ Patreon: http://www.patreon.com/numberphile Numberphile is supported by Jane Street. Learn more about them (and exciting career opportunities) at: https://bit.ly/numberphile-janestreet We are also grateful for support from the Ben Delo Foundation - https://delo.org/ NUMBERPHILE Website: http://www.numberphile.com/ Video by Brady Haran and James Hennessy Numberphile T-Shirts and Merch: https://teespring.com/stores/numberphile Brady's videos subreddit: http://www.reddit.com/r/BradyHaran/ Brady's latest videos across all channels: http://www.bradyharanblog.com/ Sign up for (occasional) emails: http://eepurl.com/YdjL9

Hannah Fry's Art at The Royal Society

1 month ago·youtube.com

+47

Proving the quadratic formula using calculus? (just my attempt)

I have been wondering if we could prove the quadratic formula using calculus, so here's what I came up with. It still involves mostly algebra but I will try to see if I can use Newton's method of doing so soon! 🛍 Shop my math t-shirts & hoodies: 👉 https://amzn.to/3qBeuw6 👉 Support this channel and get more content https://www.patreon.com/blackpenredpen #blackpenredpen #calculus

Quadratic Formula Newton's Method

1 month ago·youtube.com

+17

4211 - The Party Pooper Prime - Numberphile

OEIS founder Neil Sloane on lines of primes, awkward primes... and a 'party pooper prime'. More of this interview at https://youtu.be/u-_8wX4cECo Jane Street's puzzle at https://jane-st.co/Numberphile-JS-NumberCross4 and more puzzles at https://jane-st.co/Numberphile-JS-Puzzles More links & stuff in full description below ↓↓↓ SOME RELEVANT OEIS SEQUENCES Lines of Primes: https://oeis.org/A373813 Awkward Primes: https://oeis.org/A393445 More videos with Neil Sloane: https://www.youtube.com/playlist?list=PLt5AfwLFPxWJXQqPe_llzWmTHMPb9QvV2 Patreon: http://www.patreon.com/numberphile Numberphile is supported by Jane Street. Learn more about them (and exciting career opportunities) at: https://bit.ly/numberphile-janestreet We are also grateful for support from the Ben Delo Foundation - https://delo.org/ NUMBERPHILE Website: http://www.numberphile.com/ Videos by Brady Haran Numberphile T-Shirts and Merch: https://teespring.com/stores/numberphile Brady's videos subreddit: http://www.reddit.com/r/BradyHaran/ Brady's latest videos across all channels: http://www.bradyharanblog.com/ Thanks to ALL our eagle-eyed helpers at The Numberphile Society, especially today to Debbie, Josh and Steven

OEIS (Online Encyclopedia of Integer Sequences)Neil Sloane

1 month ago·youtube.com

+43

The Original Biggest Numbers - Numberphile

Richard Elwes discusses the original Big Numbers... long before the likes of Graham's Number and Tree(3). They are epic. More links & stuff in full description below ↓↓↓ More Richard on Numberphile: https://www.youtube.com/playlist?list=PLt5AfwLFPxWI42WXh1jnv51jHrcjatrk8 More big numbers on Numberphile: https://www.youtube.com/playlist?list=PLt5AfwLFPxWJ_FwchNAjB3xa8UnoKdmQI More about Richard Elwes (who is based at the University of Leeds): https://richardelwes.co.uk/ Richard's new book Huge Numbers Waterstones (UK): https://www.waterstones.com/book/huge-numbers/richard-elwes/9781399818827 Barnes and Noble (US): https://www.barnesandnoble.com/w/huge-numbers-richard-elwes/1147952914 Or Amazon (affiliate links)... UK: https://amzn.to/4bSZA8v US: https://amzn.to/4v3CIMp Here's the mega zoom of Patreon names: https://www.patreon.com/posts/154415033 Patreon: http://www.patreon.com/numberphile Numberphile is supported by Jane Street. Learn more about them (and exciting career opportunities) at: https://bit.ly/numberphile-janestreet We are also grateful for support from the Ben Delo Foundation - https://delo.org/ NUMBERPHILE Website: http://www.numberphile.com/ Video by Brady Haran and Pete McPartlan Numberphile T-Shirts and Merch: https://teespring.com/stores/numberphile Brady's videos subreddit: http://www.reddit.com/r/BradyHaran/ Brady's latest videos across all channels: http://www.bradyharanblog.com/ Sign up for (occasional) emails: http://eepurl.com/YdjL9

Tree(3)Graham's Number

1 month ago·youtube.com

+36

Can we have a right triangle with sides 1, x and x^2 (but the hypotenuse is x)

Just a quick, fun triangle question that I came up with a few days ago: Can we have a right triangle with sides 1, x, and x^2 BUT the hypotenuse is x? 🛍 Shop my math t-shirts & hoodies: 👉 https://amzn.to/3qBeuw6 👉 Support this channel and get more content https://www.patreon.com/blackpenredpen #blackpenredpen #calculus

Pythagorean theorem

1 month ago·youtube.com

+20

Escher's most mind-bending piece

On "The Print Gallery", by M.C. Escher Full video: https://youtu.be/ldxFjLJ3rVY

The Print Gallery

1 month ago·youtube.com

+42

So I tried integration by parts (solved another integral instead!)

I was trying to use integration by parts for the integral of 1/sqrt(x^2+16), but I ended up getting the answer for the integral of 1/(x^2+16)^(3/2) instead! Integral of 1/sqrt(x^2+16) using trig sub vs hyperbolic sub: https://youtu.be/YfrcIwdOC0U 🛍 Shop my math t-shirts & hoodies: 👉 https://amzn.to/3qBeuw6 👉 Support this channel and get more content https://www.patreon.com/blackpenredpen #blackpenredpen #calculus

Integration by Parts Trigonometric Substitution hyperbolic functions

1 month ago·youtube.com

This picture broke my brain

Escher's Print Gallery, and the tour of complex analysis it invites. Check out our virtual career fair: 3b1b.co/talent Join channel supporters to see videos early: 3b1b.co/support An equally valuable form of support is to share the videos. Home page: https://www.3blue1brown.com Original paper by de Smit and Lenstra: https://pub.math.leidenuniv.nl/~smitbde/papers/2003-de_smit-lenstra-escher.pdf Co-written by Paul Dancstep, who handled many of the animations in the art section, including the delightful mesh warp scene. Aaron Gostein helped with the manim animations in the section introducing complex functions. Artwork provided by Talia Gerhson, Mitchell Zemil, and Anna Fedczuk. Music by Vincent Rubinetti Timestamps: 0:00 - The print gallery 13:04 - Conformal maps from complex analysis 21:41 - The complex exponential 25:56 - The complex logarithm 32:32 - 3b1b Talent 33:14 - Constructing the key function 40:16 - The deeper math behind Escher ------------------ These animations are largely made using a custom Python library, manim. See the FAQ comments here: https://3b1b.co/faq#manim ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're more interested than the average viewer in lessons here. It would mean a lot to me if you chose to stay up to date on new ones, either by subscribing here on YouTube or otherwise following on whichever platform below you check most regularly. Mailing list: https://3blue1brown.substack.com Twitter: https://twitter.com/3blue1brown Bluesky: https://bsky.app/profile/3blue1brown.com Instagram: https://www.instagram.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Facebook: https://www.facebook.com/3blue1brown Patreon: https://patreon.com/3blue1brown Website: https://www.3blue1brown.com

complex analysis Escher's Print Gallery

2 months ago·youtube.com

Bacteria Grid Puzzle Solution

Part of a monthly series of puzzlers, in collaboration with MoMath and Peter Winkler

Bacteria Grid Puzzle

2 months ago·youtube.com

The Weather Equation - Numberphile

Chief Meteorologist Dan Harris discusses (at length) one of the most famous equations in weather forecasting - the Quasi Geostrophic Omega Equation. More links & stuff in full description below ↓↓↓ Catch our podcast with Dan on podcast players or here on YouTube - https://youtu.be/lnARyhaqMvU Dan works for the Met Office as a chief meteorologist - https://www.metoffice.gov.uk/ Check out some Met Office archive stuff at Brady's Objectivity channel - https://www.youtube.com/playlist?list=PLd5y2WfrtsPrauSfrdFfkE7iZ-KG9pCzO Weather videos on Brady's Sixty Symbols channel - https://www.youtube.com/playlist?list=PLcUY9vudNKBNpsv8AWUvJxbOl6S2pw8so Navier-Stokes Equation video - https://www.youtube.com/watch?v=ERBVFcutl3M Dan's website - https://www.roostweather.com/ Patreon: http://www.patreon.com/numberphile Numberphile is supported by Jane Street. Learn more about them (and exciting career opportunities) at: https://bit.ly/numberphile-janestreet We are also grateful for support from the Ben Delo Foundation - https://delo.org/ NUMBERPHILE Website: http://www.numberphile.com/ Video by Brady Haran and Pete McPartlan Numberphile T-Shirts and Merch: https://teespring.com/stores/numberphile Brady's videos subreddit: http://www.reddit.com/r/BradyHaran/ Brady's latest videos across all channels: http://www.bradyharanblog.com/ Sign up for (occasional) emails: http://eepurl.com/YdjL9

Quasi Geostrophic Omega Equation

2 months ago·youtube.com

Finding the "volume" of a 4d sphere

We will find the volume of a 4d sphere using a quadruple integral! This is called a hypersphere and has the equation x^2+y^2+z^2+w^2=R^2. And yes, there's also the hyperspherical coordinates that we could use, but I will do that in a future video. Read more on https://en.wikipedia.org/wiki/N-sphere 🛍 Shop my math t-shirts & hoodies: 👉 https://amzn.to/3qBeuw6 👉 Support this channel and get more content https://www.patreon.com/blackpenredpen #blackpenredpen #calculus

4D Sphere Volume Calculation

2 months ago·youtube.com

+18

My first triple improper integral via spherical coordinates

We will solve a fun calculus 3 triple integral of sqrt(x^2+y^2+z^2)*e^-(x^2+y^2+z^2) from -inf to inf. Make sure you know how to convert rectangular coordinates into spherical coordinates. Related videos https://youtu.be/_7Gt3Lla1pk?si=EQJ6_P7jbvncU-bU https://www.youtube.com/watch?v=uL_yq733CTY https://www.youtube.com/watch?v=u3LFnfZbkk0 🛍 Shop my math t-shirts & hoodies: 👉 https://amzn.to/3qBeuw6 👉 Support this channel and get more content https://www.patreon.com/blackpenredpen #blackpenredpen #calculus

spherical coordinates triple improper integral

2 months ago·youtube.com

+14

This calculus proof is only for math majors (National Taiwan University entrance exam)

Here's a fun calculus proof from the National Taiwan University entrance exam for transfer students. We will show that If f(0)=0 and abs(f'(x)) is less than or equal to abs(f(x)), then f(x)=0 on [0, 1/2]. The key to solving this problem is to apply the Mean Value Theorem and the Extreme Value Theorem. Try the test https://exam.lib.ntu.edu.tw/sites/default/files/exam/undergra/103/18_001-103.pdf 🛍 Shop my math t-shirts & hoodies: 👉 https://amzn.to/3qBeuw6 👉 Support this channel and get more content https://www.patreon.com/blackpenredpen #blackpenredpen #calculus

National Taiwan University entrance exam

2 months ago·youtube.com

Is x! = 0 solvable?

We know that zero factorial is equal to 1, i.e. 0!=1, but can we solve the equation x factorial is equal to 0, i.e., x!=0? We will use the regular definition of n factorial, as well as the integration extension of the factorial to see if x!=0 is possible or not. Note that although the graph of y=x! does NOT cross the x-axis, we cannot say x!=0 because of that. Think about x^2+1=0 has complex solutions. So, is there any solution? Check out can x^x=0? https://youtu.be/s4YHRtvOVB4?si=wiIPlvXtjEQOe1r5 🛍 Shop my math t-shirts & hoodies: 👉 https://amzn.to/3qBeuw6 👉 Support this channel and get more content https://www.patreon.com/blackpenredpen #blackpenredpen #calculus

Factorial Function Complex Solutions in Mathematics

2 months ago·youtube.com

+14

Volume of revolution (single integral vs triple integral)

We will find the volume of revolution of the solid generated by rotating the region bounded by y=x^2, y=9, x=0, about the y-axis. This is a typical calculus 1 problem but today we will make it 3x more fun (and get calculus 3). 🛍 Shop my math t-shirts & hoodies: 👉 https://amzn.to/3qBeuw6 👉 Support this channel and get more content https://www.patreon.com/blackpenredpen #blackpenredpen #calculus

Volume of Revolution in Calculus

2 months ago·youtube.com

A lovely calculus proof problem! (National Taiwan University entrance exam)

Let's tackle the first calculus question on a National Taiwan University transfer entrance exam. We have f and g being differential functions and d/dx(f(x))=-g(x) and d/dx(xg(x))=xf(x) and we'll show that f and g have "interlacing zeroes". Try the rest of the exam here https://www.lib.ntu.edu.tw/exam/undergra/98/98018.pdf 🛍 Shop my math t-shirts & hoodies: 👉 https://amzn.to/3qBeuw6 👉 Support this channel and get more content https://www.patreon.com/blackpenredpen #blackpenredpen #calculus

National Taiwan University entrance exam

2 months ago·youtube.com

How math majors use the intermediate value theorem!

Let f be a continuous function on [0, 1] and f(0)=f(1). Show that, for n=2, 3, 4, ..., there is some a in [0, 1-1/n] such that f(a)=f(a+1/n). Learn how to use the Intermediate Value Theorem to prove the above statement. This question appears on the transfer entrance exam at National Taiwan Univiersity: https://exam.lib.ntu.edu.tw/sites/default/files/exam/undergra/102/102018.pdf 🛍 Shop my math t-shirts & hoodies: 👉 https://amzn.to/3qBeuw6 👉 Support this channel and get more content https://www.patreon.com/blackpenredpen #blackpenredpen #calculus

Intermediate Value Theorem

2 months ago·youtube.com

+11

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